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The purpose of this work is to construct a continuous map from the homogeneous Besov space   $\dot B^0_{2,4}({\mathbb {R}}^2)$   in the set   $\mathcal {G}$   of initial data in   $\dot B^0_{2,4}({\mathbb R}^2)$   which gives birth to global solution of the mass critical non-linear Schrödinger equation in the space   $L^4({\mathbb R}^{1+2})$  . We use the fact that solutions of scale which are different enough almost do not interact; the main point is that we determine a condition about the size of the scale which depends continuously on the data.

From an Initial Data to a Global Solution of the Non-linear Schrödinger Equation: A Building Process